Uniform Approximations of Bernoulli and Euler Polynomials in Terms of Hyperbolic Functions

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for B _n ( nz +1/2) and E _n ( nz +1/2) in powers of n ⁻¹, and coefficients are rational functions of z and hyperbolic functions of argument 1/(2 z ). These expansions are uniformly valid for | z ± i /2π|>1/2π and | z ± i /π|1/π, respectively. For a real argument, the accuracy of these approximations is restricted to the monotonic region. The range of validity of the uniformity parameter z is enlarged, respectively, to regions of the form | z ± i /2( m +1)π|>1/2( m +1)π and | z ± i /(2 m +1)π|>1/(2 m +1)π, m =1,2,3,…, by adding certain combinations of incomplete gamma functions to these uniform expansions. In addition, the convergence of these improved expansions is stronger, and for a real argument, the accuracy of these improved approximations is also better in the oscillatory region.